Sure, I can help with that! Let's simplify the expression step-by-step:
Given expression: [tex]\( 7 \sqrt{7} - 9 \sqrt{5} + 6 \sqrt{5} + 20 \sqrt{7} \)[/tex]
1) Group the like terms together. In this case, combine the terms involving [tex]\( \sqrt{7} \)[/tex] and the terms involving [tex]\( \sqrt{5} \)[/tex]:
[tex]\[ (7 \sqrt{7} + 20 \sqrt{7}) + (-9 \sqrt{5} + 6 \sqrt{5}) \][/tex]
2) Combine the coefficients of [tex]\( \sqrt{7} \)[/tex]:
[tex]\[ 7 \sqrt{7} + 20 \sqrt{7} = (7 + 20) \sqrt{7} = 27 \sqrt{7} \][/tex]
3) Similarly, combine the coefficients of [tex]\( \sqrt{5} \)[/tex]:
[tex]\[ -9 \sqrt{5} + 6 \sqrt{5} = (-9 + 6) \sqrt{5} = -3 \sqrt{5} \][/tex]
4) Now, put these simplified terms together:
[tex]\[ 27 \sqrt{7} - 3 \sqrt{5} \][/tex]
Next, let's find the numerical values:
5) Calculate [tex]\( 27 \sqrt{7} \)[/tex]:
[tex]\[ 27 \sqrt{7} \approx 27 \times 2.6457513110645906 \approx 71.43528539874394 \][/tex]
6) Calculate [tex]\( -3 \sqrt{5} \)[/tex]:
[tex]\[ -3 \sqrt{5} \approx -3 \times 2.23606797749979 \approx -6.708203932499369 \][/tex]
7) Combine these numerical results:
[tex]\[ 71.43528539874394 - 6.708203932499369 \approx 64.72708146624457 \][/tex]
So, the simplified expression [tex]\( 7 \sqrt{7} - 9 \sqrt{5} + 6 \sqrt{5} + 20 \sqrt{7} \)[/tex] evaluates to approximately [tex]\( 64.72708146624457 \)[/tex].
